On Bounds of k -Fractional Integral Operators with Mittag-Lefﬂer Kernels for Several Types of Exponentially Convexities

: This paper aims to study the bounds of k -integral operators with the Mittag-Lefﬂer kernel in a uniﬁed form. To achieve these bounds, the deﬁnition of exponentially ( α , h − m ) − p -convexity is utilized frequently. In addition, a fractional Hadamard type inequality which shows the upper and lower bounds of k -integral operators simultaneously is presented. The results are directly linked with the results of many published articles.


Introduction
Special functions, including trigonometric, hyperbolic, exponential, gamma, beta, and many others, have fascinating and unique characteristics. They play very important role in the fields of mathematical analysis, complex analysis, geometric function theory, physics, and statistics. The well known Mittag-Leffler function introduced in [1] represents a vital contribution to the class of special functions. It is very frequently used in applied sciences in regard to the generalization and extension of classical concepts; for further details, readers are referred to [2][3][4][5].
The Mittag-Leffler function is frequently utilized in the formation of generalizations of fractional integral operators. Fractional integral operators lead to the theory of fractional calculus, fractional analysis, fractional differential equations, and fractional dynamic systems; see [6][7][8]. The aim of this paper is to estimate fractional integral operators containing a specific Mittag-Leffler function via various types of exponential convexities.
The Mittag-Leffler function is a generalization of exponential, trigonometric, and hyperbolic functions, and is defined with the help of the gamma function. Likewise, the beta function can be utilized to extend the Mittag-Leffler function. In the following, we provide the definitions of the gamma function, beta function, -beta function, and pochhammer symbol. Definition 1 ([5]). The gamma function for Φ > 0 is defined by: (1) Definition 2 ([5]). The beta function is defined by: Definition 3 ([9]). The definition of the -beta function is defined by: where min{ (Ψ), (Φ)} > 0 and R( ) > 0.
From k-I Os (4) and (5), for constant function we can write: Next, we provide the definition of newly defined functions, namely, exponentially (α, h − m) − p-convex functions, as follows: is valid, while J ⊆ R is an interval involving (0, 1) and h : J → R is a positive function with In recent years, many authors have derived the bounds of several IOs for different kinds of convex functions. For example, Farid [28] established the bounds of Riemann-Liouville IOs using convex functions. Mehmood and Farid [29] provided the bounds of generalized Riemann-Liouville k-I Os via m-convex functions. Yu et al. [30] proved the bounds of generalized IOs involving the Mittag-Leffler function in their kernels via strongly exponentially (α, h − m)-convex functions. Inspired by these previous works, the aim of this paper is to derive the bounds of generalized k-I Os for exponentially (α, h − m) − p-convex functions.
In the upcoming section, we first derive the bounds of the k-I Os provided in (4) and (5) for functions satisfying (9) and derive a modulus inequality for these operators. Further, an identity is proved to derive the Hadamard type inequality for k-I Os via exponentially (α, h − m) − p-convex functions. In particular cases, the presented results provide bounds of various IOs.

Main Results
First, we provide the bounds of k-I Os via exponentially (α, h − m) − p-convex functions: In addition, let Z be differentiable and strictly increasing with Z ∈ L 1 [µ, ν]. Then, for η, ζ ≥ k and ω ∈ R, we have: Proof. Under the given assumptions, the following inequalities hold: By utilizing the exponentially (α, h − m) − p-convexity of Y, we can obtain the following: From inequalities (11) and (13), the following inequality is valid: By utilizing the integral operator (4) on the left-hand side and making the substitution R = ( − δ)/( − µ) on the right-hand side, we obtain: The above inequality can be written in the following form: On the other hand, by multiplying (12) and (14) and adopting the same approach as we did for (11) and (13), the following inequality can be obtained: The above inequality takes the following form: By adding the inequalities (15) and (16), the inequality (10) is obtained. (10), the following inequality is valid: In the following, we provide the modulus inequality for k-I Os via exponentially (α, h − m) − p-convex functions: Theorem 2. Let Y : [µ, ν] −→ R be positive integrable and let |Y | be an exponentially (α, h − m) − p-convex function with m ∈ (0, 1]. In addition, let Z be differentiable and strictly increasing with Z ∈ L 1 [µ, ν]. Then, for η, ζ ≥ k and ω ∈ R, we have: Proof. By utilizing the strongly exponentially (α, h − m) − p-convexity of |Y |, the following inequality holds: The above inequality can be written in the following form: Now, by multiplying the inequality (11) with the first inequality of (20) and integrating over [µ, ], we obtain the following: After simplifying the inequality (21), we have By using the second inequality of (20) and following the same approach as we did for the first inequality, we can obtain the following: From inequalities (22) and (23), we have: Again, by utilizing the strongly exponentially (α, h − m) − p-convexity of |Y |, we have By following the same approach as we did for (11) and (19), from (12) and (25) we can obtain: By adding the inequalities (24) and (26), the required inequality (18) is obtained.

Remark 4.
From Theorem 2, the bounds for all IOs (provided in Remark 1) via all kinds of convex functions (provided in Remark 2) can be obtained.
The following identity is useful to prove the Hadamard-type inequality: holds for m ∈ (0, 1], then we have Proof. We use the following identity: By applying the exponentially (α, h − m) − p-convexity of Y, we obtain: By using the assumption provided in (28), the required inequality (29) is obtained.
In the following, we provide the Hadamarad type inequality for k-I Os via exponentially (α, h − m) − p-convex functions: Theorem 3. With the same conditions on Y, Z, and h as in Theorem 1, and additionally if (28) is satisfied, then we have: where U (δ) = δ 1 p and Q(ω) = e ωµ for ω ≥ 0; Q(ω) = e ων for ω < 0.

Corollary 3.
For η = ζ in (32), the following inequality is valid: Remark 5. From Theorem 3, Hadamard-type inequalities for all kinds of IOs (provided in Remark 1) for all kinds of convex functions (provided in Remark 2) can be obtained.

Concluding Remarks
In this article, we have investigated the bounds of k-I Os. These bounds were achieved by applying thedefinition of exponentially (α, h − m) − p-convex functions. The presented results provide a large number of new bounds of several IOs for various kinds of convexities using convenient substitutions. Further, an identity is established to prove the Hadamard-type inequality for k-I Os via exponentially (α, h − m) − p-convex functions.